Index: The Book of Statistical ProofsStatistical Models ▷ Frequency data ▷ Dirichlet-distributed data ▷ Maximum likelihood estimation

Theorem: Let there be a Dirichlet-distributed data set $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$:

$\label{eq:Dir} y_i \sim \mathrm{Dir}(\alpha), \quad i = 1, \ldots, n \; .$

Then, the maximum likelihood estimate for the concentration parameters $\alpha$ can be obtained by iteratively computing

$\label{eq:Dir-MLE} \alpha_j^{\text{(new)}} = \psi^{-1}\left[ \psi\left( \sum_{j=1}^k \alpha_j^{\text{(old)}} \right) + \log \bar{y}_j \right]$

where $\psi(x)$ is the digamma function and $\log \bar{y}_j$ is given by:

$\label{eq:log-pi} \log \bar{y}_j = \frac{1}{n} \sum_{i=1}^n \log y_{ij} \; .$

Proof: The likelihood function for each observation is given by the probability density function of the Dirichlet distribution

$\label{eq:Dir-yi} p(y_i|\alpha) = \frac{\Gamma\left( \sum_{j=1}^k \alpha_j \right)}{\prod_{j=1}^k \Gamma(\alpha_j)} \, \prod_{j=1}^k {y_{ij}}^{\alpha_j-1}$

and because observations are independent, the likelihood function for all observations is the product of the individual ones:

$\label{eq:Dir-LF} p(y|\alpha) = \prod_{i=1}^n p(y_i|\alpha) = \prod_{i=1}^n \left[ \frac{\Gamma\left( \sum_{j=1}^k \alpha_j \right)}{\prod_{j=1}^k \Gamma(\alpha_j)} \, \prod_{j=1}^k {y_{ij}}^{\alpha_j-1} \right] \; .$

Thus, the log-likelihood function is

$\label{eq:Dir-LL} \mathrm{LL}(\alpha) = \log p(y|\alpha) = \log \prod_{i=1}^n \left[ \frac{\Gamma\left( \sum_{j=1}^k \alpha_j \right)}{\prod_{j=1}^k \Gamma(\alpha_j)} \, \prod_{j=1}^k {y_{ij}}^{\alpha_j-1} \right]$

which can be developed into

$\label{eq:Dir-LL-der} \begin{split} \mathrm{LL}(\alpha) &= \sum_{i=1}^n \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) - \sum_{i=1}^n \sum_{j=1}^k \log \Gamma(\alpha_j) + \sum_{i=1}^n \sum_{j=1}^k (\alpha_j-1) \log y_{ij} \\ &= n \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) - n \sum_{j=1}^k \log \Gamma(\alpha_j) + n \sum_{j=1}^k (\alpha_j-1) \, \frac{1}{n} \sum_{i=1}^n \log y_{ij} \\ &= n \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) - n \sum_{j=1}^k \log \Gamma(\alpha_j) + n \sum_{j=1}^k (\alpha_j-1) \, \log \bar{y}_j \end{split}$

where we have specified

$\label{eq:log-pi-reit} \log \bar{y}_j = \frac{1}{n} \sum_{i=1}^n \log y_{ij} \; .$

The derivative of the log-likelihood with respect to a particular parameter $\alpha_j$ is

$\label{eq:Dir-dLLdaj} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\alpha)}{\mathrm{d}\alpha_j} &= \frac{\mathrm{d}}{\mathrm{d}\alpha_j} \left[ n \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) - n \sum_{j=1}^k \log \Gamma(\alpha_j) + n \sum_{j=1}^k (\alpha_j-1) \, \log \bar{y}_j \right] \\ &= \frac{\mathrm{d}}{\mathrm{d}\alpha_j} \left[ n \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) \right] - \frac{\mathrm{d}}{\mathrm{d}\alpha_j} \left[ n \log \Gamma(\alpha_j) \right] + \frac{\mathrm{d}}{\mathrm{d}\alpha_j} \left[ n (\alpha_j-1) \, \log \bar{y}_j \right] \\ &= n \psi\left( \sum_{j=1}^k \alpha_j \right) - n \psi(\alpha_j) + n \log \bar{y}_j \end{split}$

where we have used the digamma function

$\label{eq:digamma-fct} \psi(x) = \frac{\mathrm{d}\log \Gamma(x)}{\mathrm{d}x} \; .$

Setting this derivative to zero, we obtain:

$\label{eq:Dir-dLLdaj-0} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\alpha)}{\mathrm{d}\alpha_j} &= 0 \\ 0 &= n \psi\left( \sum_{j=1}^k \alpha_j \right) - n \psi(\alpha_j) + n \log \bar{y}_j \\ 0 &= \psi\left( \sum_{j=1}^k \alpha_j \right) - \psi(\alpha_j) + \log \bar{y}_j \\ \psi(\alpha_j) &= \psi\left( \sum_{j=1}^k \alpha_j \right) + \log \bar{y}_j \\ \alpha_j &= \psi^{-1} \left[ \psi\left( \sum_{j=1}^k \alpha_j \right) + \log \bar{y}_j \right] \; . \end{split}$

In the following, we will use a fixed-point iteration to maximize $\mathrm{LL}(\alpha)$. Given an initial guess for $\alpha$, we construct a lower bound on the likelihood function \eqref{eq:Dir-LL-der} which is tight at $\alpha$. The maximum of this bound is computed and it becomes the new guess. Because the Dirichlet distribution belongs to the exponential family, the log-likelihood function is convex in $\alpha$ ánd the maximum is the only stationary point, such that the procedure is guaranteed to converge to the maximum.

In our case, we use a bound on the gamma function

$\label{eq:gamma-fct-bound} \begin{split} \Gamma(x) &\geq \Gamma(\hat{x}) \cdot \mathrm{exp}\left[(x-\hat{x}) \, \psi(\hat{x}) \right] \\ \log \Gamma(x) &\geq \log \Gamma(\hat{x}) + (x-\hat{x}) \, \psi(\hat{x}) \end{split}$

and apply it to $\Gamma\left( \sum_{j=1}^k \alpha_j \right)$ in \eqref{eq:Dir-LL-der} to yield

$\label{eq:Dir-LL-bound} \begin{split} \frac{1}{n} \mathrm{LL}(\alpha) &= \log \Gamma\left( \sum_{j=1}^k \alpha_j \right) - \sum_{j=1}^k \log \Gamma(\alpha_j) + \sum_{j=1}^k (\alpha_j-1) \, \log \bar{y}_j \\ \frac{1}{n} \mathrm{LL}(\alpha) &\geq \log \Gamma\left(\sum_{j=1}^k \hat{\alpha}_j\right) + \left(\sum_{j=1}^k \alpha_j - \sum_{j=1}^k \hat{\alpha}_j\right) \psi\left(\sum_{j=1}^k \hat{\alpha}_j\right) - \sum_{j=1}^k \log \Gamma(\alpha_j) + \sum_{j=1}^k (\alpha_j-1) \, \log \bar{y}_j \\ \frac{1}{n} \mathrm{LL}(\alpha) &\geq \left(\sum_{j=1}^k \alpha_j\right) \psi\left(\sum_{j=1}^k \hat{\alpha}_j\right) - \sum_{j=1}^k \log \Gamma(\alpha_j) + \sum_{j=1}^k (\alpha_j-1) \, \log \bar{y}_j + \mathrm{const.} \end{split}$

which leads to the following fixed-point iteration using \eqref{eq:Dir-dLLdaj-0}:

$\label{eq:Dir-MLE-qed} \alpha_j^{\text{(new)}} = \psi^{-1}\left[ \psi\left( \sum_{j=1}^k \alpha_j^{\text{(old)}} \right) + \log \bar{y}_j \right] \; .$
Sources:

Metadata: ID: P182 | shortcut: dir-mle | author: JoramSoch | date: 2020-10-22, 09:31.